Day Lab-3: Godzilla Geometry a.k.a. Distance Ladder - KICP

Day Lab-3: Godzilla Geometry a.k.a. Distance Ladder(Randy Landsberg, Mark SubbaRao, & Takemi Okamoto)The further away something is the smaller itappears. Think of the skyscrapers in the Loop thatmake up the Chicago skyline. From five or so milesaway in Hyde Park, enormous structures like theHancock Building, one of the tallest buildings in theworld, appear smaller than modest local landmarks.From the observation deck on the 93 rd floor of theHancock Building the people at street level look like ants. Our brains help us to make sense ofthese illusions, movie makers use them for special effects, and scientists take advantage of themto determine distances to far away objects.GODZILLA (1954 - 1975) / aka: GojiraHeight: 50 meters (164 feet)Mass: 20,000 metric tons (22,000 tons)Fight record: 18 wins, 3 losses, 7 tiesPowers/weapons: Atomic ray, superregenerative power (Godzilla can bewounded, but his G-cells heal veryrapidly)Year of first appearance: 1954, inGodzillaOrigin: Monster brought to life by Atomictesting in the Pacific Ocean during the '50's. source: http://www.avia-art.com/godzillawebsite/ginfo.htmlIt is possible to use this effect to measure distance because the apparent change in size of anobject is related to how far away that object is and can be well described with mathematics. Thereare some rules to follow (e.g., you need to look at the object the same way each time). In thislaboratory we will empirically determine how apparent size changes with distance for everydayobjects (and movie monsters) on the grounds of Yerkes. We will then use our measurements, in away very similar to how astronomers do, to indirectly determine distances (and relative sizes).This laboratory and Night Lab-3, the Distance Ladder, are meant to be complementary. Theyboth demonstrate that one can determine distances and relative sizes, without having to actuallytravel to the object being measured, and build on earlier measurements to develop a distanceladder that takes one further and further beyond one’s reach.Crossing the Digital Divide: Connecting Pixels to AnglesIn this laboratory we will use a digital camera, specifically a Nikon Coolpix 4500. When a digitalcamera takes a picture, the image is divided up into a grid of tiny identically sized and shapedpixels or picture elements. Digital cameras naturally divide images into lots of tiny pixels becauselight coming through the camera’s lens falls on a charge-coupled device (CCD), which is physicallydivided into many pixels. The CCD chip in the Nikon 4500 has 4 mega pixels, or 4 million pixels.The dimensions of the CCD are 2,272 pixels wide by 1,704 pixels tall (note 2,272 x 1,704 =3,871,488 is rounded up to 4 million by optimistic advertising).Astronomers like to measure objects in terms of angular scale (most often very very small angular2003 Yerkes Summer Institute Day Lab 3 – Godzilla Geometry page-12

scales); the objects they observe are very very very far away. Angular size is a very usefulmeasurement because one does not need to know exact distances or sizes to make usefulmeasurements, and because it is easy to compare images from one telescope to another.Heighthf 1f 2Distance near d1Distance far d2Figure: Side view of same height object observed from near and far. Note: when the object isnear the angular size (f1) is a much bigger angle than the apparent angular size when it is far (f2)A bigger angular size translates to a bigger image hitting more pixels. So the object of height h willtake up more pixels in the near image (corresponding to angular size f1) than in the far awaypicture (with smaller angular size, f2). If the camera set-up is kept constant, then the number ofpixels and the angular size are directly proportional, to a very good approximation:f x constant = number of pixels.It will be simpler for us to determine the relationship between distance and the apparent size of anobject measured in number of pixels, rather than in degrees.Fortunately, in many astronomical situations the object is so distant and so tiny that we can use ashortcut called the small angle approximation. When the conditions are right to use the smallangle approximation size is directly proportional to distance. That is, if something looks half as big,it is twice as far (assuming that the something is either same object or two objects that are thesame size). We cannot assume that the small angle approximation will work for the distances andobjects that we will be working with. Instead we will measure the relationship of apparent angularsize to distance.For the extra curious:The vertical angular field of view of the Nikon Coolpix 4500 can be determined by takingphotos of an object of a known height from a known distance. We now know two sides of aright triangle. Trigonometry tells us that if we know the length of the opposite and adjacentsides of a right triangle then we can find the size of the angletan (f) = opposite/adjacentfor example for the figure above tan (f1) = h/d1, sof(radians) = arctan (opposite/adjacent)and f1 = arctan (h/d1)f(degrees)= f(radians)x180/p f1(degrees)= {arctan(h/d1)} x 180/pNow we know the angular scale of the object. By counting the number of pixels that it takesup in the image we now know the size in terms of angular scale and number of pixels.We can equate them: (angular size f) = (number of pixels)and divide to find the conversion factor: #pixels/degree.2003 Yerkes Summer Institute Day Lab 3 – Godzilla Geometry page-13

Under the conditions that this lab will use (e.g., focus set at infinity, zoom all the way in) wehave determined that the vertical angular field of view of the Nikon 4500 is about (45degrees). This means that it takes about 38 of the 1,704 vertical pixels to correspond to anangular size of one degree. (e.g., expect about 19 vertical pixels for a full moon).Determine How Apparent Size Changes With Distance:In this part of the lab you will take digital pictures of a golf flag and a model Godzilla, objects ofknown heights, from different distances and measure how their apparent sizes change.Hardware: Digital Camera, (LCD projector), long tape measure, standard objects (golf flag &model Godzilla of known heights), dark ground cover for contrast, computers with pcmica slot,HOU software and JPG to FITS file converter.IMPORTANT: it is critical that the camera is set up the same way each time otherwise theauto zoom will change the perspective. The camera should be set to infinity (mountainpeak icon) and the zoom all the way in (i.e. no zoom). The camera should also be mountedin a fixed fashion to the tripod.Step 1 – Measure the heights of Godzilla & the Golf Flag. Record in your lab notebook.Step 2 - Take photos of Godzilla and the golf flag from different distances (from ~10 to 300meters)Your group will need to develop a system to:a) To determine the distance andb) To index the photos (e.g., which photo is from which distance)Step 3 – Analyze your dataYour lab instructors will demonstrate how to download the images on to the computers and thenhow to use the software:a) To convert the files from JPG to FITS format andb) To determine how many pixels tall each object is• Measure the height of Godzilla and the golf flag for each distance and record thesevalues in your lab notebook.• Create a data table for both objects (distance, number of pixels).• Record in your notebook any relevant details of how you determined where Godzillaor the golf flag began and ended.Step 4 – Graph your data• Generate a graph of apparent size (pixels) vs. distance (meters).• Comment in your lab notebook on anything you observe in this graph.Next you will use the results of your Godzilla/Golf Flag apparent size vs. distance measurementsto determine the distances to far away objects and their sizes. As a group you will pick challengesand agree on experiments to solve them. Record your method, observations and results in yourlab notebook.2003 Yerkes Summer Institute Day Lab 3 – Godzilla Geometry page-14

Distance & Size Challenge – Distance to the Dome? Height of the Dome?Use a model Godzilla of known height, and the distance/apparent size relationship developedearlier to determine how far away from the big dome you are at any point on the lawn and todetermine how tall the dome is. Then measure this distance directly and compare your results.Compare your height estimate to the know height.Distance Challenge– How far to that flag?Determine the distance to a far away golf flag on the George Williams golf course using thedistance vs. angular size relationship developed earlier. Then measure this distance directly(when no one is golfing) and compare your results. Assume all golf flags are the same height.Monster Movie Challenge – How to make the Model Godzilla appear as big as the dome inthe camera?Use the apparent size/distance relationship and the height of the dome to estimate how to positionthe camera, the model and the dome to create a perspective that makes Godzilla appear about thesame height as the dome. Test your prediction with the camera. Are you ready for Hollywood?Image ChallengeClimb up the cosmological distance ladder by comparing the sizes of similarextragalactic objects in images from the Sloan Digital Sky Survey.First, based on the assumption that globular clusters belonging to the Milky Way and M31 are ofcomparable size (or luminosity), determine the relative distance to M31.Next, assuming that the galaxies in a distant cluster are like M31 in size, find the relative distanceto that cluster.Finally, translate these observations into absolute distances - you will be given the distance to theoriginal globular cluster.Some Useful Conversion Factors1 meter = 3.2808 feet360° in a circle1° = 60 arcminutes = 60’1 arcminute = 1’ = 60 arcseconds = 60”sothere are 60(arcmin/deg)x60(arcsec/arcmin) = 3,600 arcseconds per degree1 arcsecond = 1” = 1/3600°2003 Yerkes Summer Institute Day Lab 3 – Godzilla Geometry page-15